Higher Genus FJRW Theory for Fermat Cubic Singularity

Springer Science and Business Media LLC - Tập 37 - Trang 1179-1204 - 2021
Xin Wang1
1School of Mathematics, Shandong University, Jinan, P.R. China

Tóm tắt

In this paper, we study the higher genus FJRW theory of Fermat cubic singularity with maximal group of diagonal symmetries using Givental formalism. As results, we prove the finite generation property and holomorphic anomaly equation for the associated FJRW theory. Via general LG-LG mirror theorem, our results also hold for the Saito-Givental theory of the Fermat cubic singularity.

Tài liệu tham khảo

Berglund, P., Henningson, M.: Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus. Nuclear Physics B, 433(2), 311–332 (1995) Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nuclear Physics B, 393(1–2), 377–391 (1993) Buryak, A., Janda, F., Pandharipande, R.: The hypergeometric functions of the Faber-Zagier and Pixton relations. Pure and Applied Mathematics Quarterly, 11(4), 591–631 (2015) Etingof, P.: Mathematical ideas and notions of quantum field theory. Available at http://www-math.mit.edu/etingof/lect.ps (2002) Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry, and quantum singularity theory. Annals of Mathematics, 1–106 (2013) Givental, A. B.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Moscow Mathematical Journal, 1(4), 551–568 (2001) Givental, A. B.: Semisimple Frobenius structures at higher genus. International Mathematics Research notices, 23, 1265–1286 (2001) Givental, A. B.: Symplectic geometry of Frobenius structures. In: Frobenius Manifolds, Friedr. Vieweg, Wiesbaden, 91–112 (2004) Guo, S., Janda, F., Ruan, Y.: Structure of higher genus Gromov-Witten invariants of quintic 3-folds. arXiv:1812.11908 (2018) He, W., Li, S., Shen, Y., et al.: Landau-Ginzburg mirror symmetry conjecture. arXiv:1503.01757 (2015) Iritani, H., Milanov, T., Ruan, Y., et al.: Gromov-Witten theory of quotients of Fermat Calabi-Yau varieties. Mem. Amer. Math. Soc., 269(1310), (2021) Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. arXiv:0906.0796 (2009) Krawitz, M. and Shen, Y.: Landau-Ginzburg/Calabi-Yau correspondence of all genera for elliptic orbifold ℙ1. arXiv:1106.6270 (2011) Lho, H. and Pandharipande, R.: Stable quotients and the holomorphic anomaly equation. Advances in Mathematics, 332, 349–402 (2018) Li, C., Li, S. and Saito, K.: Primitive forms via polyvector fields. arXiv:1311.1659 (2013) Milanov, T.: Analyticity of the total ancestor potential in singularity theory. Advances in Mathematics, 255, 217–241 (2014) Noumi, M. and Yamada, Y.: Notes on the flat structures associated with simple and simply elliptic singularities. In: Integrable Systems and Algebraic Geometry, (Kobe/Kyoto 1997), World Sci. Publ., River Edge, NJ, 373–383 (1998) Pandharipande, R., Pixton, A. and Zvonkine, D.: Relations on \({\overline M _{g,n}}\) via 3-spin structures. Journal of the American Mathematical Society, 28(1), 279–309 (2015) Saito, K.: Quasihomogene isolierte singularitäten von hyperflächen. Inventiones Mathematicae, 14(2), 123–142 (1971) Saito, K.: Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3), 775–792 (1981) Saito, K.: The higher residue pairings KF(k) for a family of hypersurface singular points. In: Singularities, Proc. Sympos. Pure Math, Vol. 40, Amer. Math. Soc., Providence, RI, 441–463 (1983) Saito, K.: Period mapping associated to a primitive form. Publications of the Research Institute for Mathematical Sciences, 19(3), 1231–1264 (1983) Strachan, I. A.: Simple elliptic singularities: a note on their G-function. Asian Journal of Mathematics, 16(3), 409–426 (2012) Teleman, C.: The structure of 2d semi-simple field theories. Inventiones Mathematicae, 188(3), 525–588 (2012) Wang, X.: Finite generation and holomorphic anomaly equation for equivariant Gromov-Witten invariants of \({K_{{^1} \times {^1}}}\). arXiv:1908.03691 (2019) Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. Topological Methods in Modern Mathematics, 235 (1993)